EXAMPLE 2 Multiply rational expressions involving polynomials Find the product 3x 2 + 3x 4x 2 – 24x + 36 x 2 – 4x + 3 x 2 – x Multiply numerators and denominators.

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EXAMPLE 2 Multiply rational expressions involving polynomials Find the product 3x 2 + 3x 4x 2 – 24x + 36 x 2 – 4x + 3 x 2 – x Multiply numerators and denominators. 3x 2 + 3x 4x 2 – 24x + 36 x 2 – 4x + 3 x 2 – x = (3x 2 + 3x) (x 2 – 4x + 3) (4x 2 – 24x + 36)(x 2 – x) Factor and divide out common factors. = 3(x + 1) 4(x – 3) Simplify. = 3x(x + 1)(x – 3)(x – 1) 4x(x – 3)(x – 3)(x – 1)

EXAMPLE 2 Multiply rational expressions involving polynomials CHECK Check your simplification using a graphing calculator. The graphs coincide. So, the expressions are equivalent for all values of x other than the excluded values (0, 1, and 3). Graph y 1 = 3x 2 + 3x 4x 2 – 24x + 36 x 2 – 4x + 3 x 2 – x and y 2 = 3(x + 1) 4(x – 3).

EXAMPLE 3 Multiply a rational expression by a polynomial Find the product 5x5x x 2 + 5x + 6 (x + 3). 5x5x x 2 + 5x + 6 (x + 3). 5x5x x 2 + 5x + 6 (x + 3) = 1 Rewrite polynomial as a fraction. = 5x(x + 3) x 2 + 5x + 6 Multiply numerators and denominators. = 5x(x + 3) (x + 3)(x + 2) Factor and divide out common factor. Simplify. = 5x5x x + 2

GUIDED PRACTICE for Examples 2 and 3 3. Find the product x 2 + x – 2 x 2 + x 2x 2 + 2x 5x 2 –15x +10 2(x + 1) 5(x – 2) ANSWER Find the product 2w22w2 w 2 – 7w + 12 (w – 4). 4. 2w22w2 w – 3 ANSWER

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